%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : GRP057-1 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 10:09:10 AM UTC 2026
% Result : Unsatisfiable 0.62s 0.58s
% Output : Proof 2.06s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : GRP057-1 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n002.cluster.edu
% 0.09/0.36 % Model : x86_64 x86_64
% 0.09/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.36 % Memory : 8046.5625MB
% 0.09/0.36 % OS : Linux 6.8.0-71-generic
% 0.09/0.36 % CPULimit : 300
% 0.09/0.37 % WCLimit : 300
% 0.09/0.37 % DateTime : Sun Sep 27 09:19:51 UTC 2026
% 0.15/0.37 % CPUTime :
% 0.15/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.62/0.58 Command-line arguments: --no-flatten-goal
% 0.62/0.58
% 0.62/0.58 % SZS status Unsatisfiable
% 0.62/0.58
% 2.06/0.73 % SZS output start Proof
% 2.06/0.73 Axiom 1 (single_axiom): multiply(X, inverse(multiply(multiply(inverse(multiply(inverse(Y), multiply(inverse(X), Z))), W), inverse(multiply(Y, W))))) = Z.
% 2.06/0.73
% 2.06/0.73 Lemma 2: inverse(multiply(multiply(inverse(multiply(inverse(X), multiply(inverse(inverse(Y)), Z))), W), inverse(multiply(X, W)))) = multiply(Y, inverse(multiply(multiply(inverse(multiply(inverse(V), Z)), U), inverse(multiply(V, U))))).
% 2.06/0.73 Proof:
% 2.06/0.73 inverse(multiply(multiply(inverse(multiply(inverse(X), multiply(inverse(inverse(Y)), Z))), W), inverse(multiply(X, W))))
% 2.06/0.73 = { by axiom 1 (single_axiom) R->L }
% 2.06/0.73 multiply(Y, inverse(multiply(multiply(inverse(multiply(inverse(V), multiply(inverse(Y), inverse(multiply(multiply(inverse(multiply(inverse(X), multiply(inverse(inverse(Y)), Z))), W), inverse(multiply(X, W))))))), U), inverse(multiply(V, U)))))
% 2.06/0.73 = { by axiom 1 (single_axiom) }
% 2.06/0.73 multiply(Y, inverse(multiply(multiply(inverse(multiply(inverse(V), Z)), U), inverse(multiply(V, U)))))
% 2.06/0.73
% 2.06/0.73 Lemma 3: multiply(inverse(X), multiply(X, inverse(multiply(multiply(inverse(multiply(inverse(Y), Z)), W), inverse(multiply(Y, W)))))) = Z.
% 2.06/0.73 Proof:
% 2.06/0.73 multiply(inverse(X), multiply(X, inverse(multiply(multiply(inverse(multiply(inverse(Y), Z)), W), inverse(multiply(Y, W))))))
% 2.06/0.73 = { by lemma 2 R->L }
% 2.06/0.73 multiply(inverse(X), inverse(multiply(multiply(inverse(multiply(inverse(V), multiply(inverse(inverse(X)), Z))), U), inverse(multiply(V, U)))))
% 2.06/0.73 = { by axiom 1 (single_axiom) }
% 2.06/0.73 Z
% 2.06/0.73
% 2.06/0.73 Lemma 4: inverse(multiply(multiply(inverse(multiply(inverse(X), Y)), Z), inverse(multiply(X, Z)))) = multiply(W, inverse(multiply(multiply(inverse(Y), V), inverse(multiply(inverse(W), V))))).
% 2.06/0.73 Proof:
% 2.06/0.73 inverse(multiply(multiply(inverse(multiply(inverse(X), Y)), Z), inverse(multiply(X, Z))))
% 2.06/0.73 = { by axiom 1 (single_axiom) R->L }
% 2.06/0.73 multiply(W, inverse(multiply(multiply(inverse(multiply(inverse(inverse(W)), multiply(inverse(W), inverse(multiply(multiply(inverse(multiply(inverse(X), Y)), Z), inverse(multiply(X, Z))))))), V), inverse(multiply(inverse(W), V)))))
% 2.06/0.73 = { by lemma 3 }
% 2.06/0.73 multiply(W, inverse(multiply(multiply(inverse(Y), V), inverse(multiply(inverse(W), V)))))
% 2.06/0.73
% 2.06/0.73 Lemma 5: multiply(inverse(X), multiply(X, multiply(Y, inverse(multiply(multiply(inverse(Z), W), inverse(multiply(inverse(Y), W))))))) = Z.
% 2.06/0.73 Proof:
% 2.06/0.73 multiply(inverse(X), multiply(X, multiply(Y, inverse(multiply(multiply(inverse(Z), W), inverse(multiply(inverse(Y), W)))))))
% 2.06/0.73 = { by lemma 4 R->L }
% 2.06/0.73 multiply(inverse(X), multiply(X, inverse(multiply(multiply(inverse(multiply(inverse(V), Z)), U), inverse(multiply(V, U))))))
% 2.06/0.73 = { by lemma 3 }
% 2.06/0.73 Z
% 2.06/0.73
% 2.06/0.73 Lemma 6: multiply(inverse(Z), multiply(Z, Y)) = multiply(inverse(X), multiply(X, Y)).
% 2.06/0.73 Proof:
% 2.06/0.73 multiply(inverse(Z), multiply(Z, Y))
% 2.06/0.73 = { by axiom 1 (single_axiom) R->L }
% 2.06/0.73 multiply(inverse(Z), multiply(Z, multiply(W, inverse(multiply(multiply(inverse(multiply(inverse(inverse(W)), multiply(inverse(W), Y))), U), inverse(multiply(inverse(W), U)))))))
% 2.06/0.73 = { by lemma 5 }
% 2.06/0.73 multiply(inverse(inverse(W)), multiply(inverse(W), Y))
% 2.06/0.73 = { by lemma 5 R->L }
% 2.06/0.73 multiply(inverse(X), multiply(X, multiply(W, inverse(multiply(multiply(inverse(multiply(inverse(inverse(W)), multiply(inverse(W), Y))), V), inverse(multiply(inverse(W), V)))))))
% 2.06/0.73 = { by axiom 1 (single_axiom) }
% 2.06/0.73 multiply(inverse(X), multiply(X, Y))
% 2.06/0.73
% 2.06/0.73 Lemma 7: multiply(multiply(inverse(X), W), inverse(multiply(Z, W))) = multiply(multiply(inverse(X), Y), inverse(multiply(Z, Y))).
% 2.06/0.73 Proof:
% 2.06/0.73 multiply(multiply(inverse(X), W), inverse(multiply(Z, W)))
% 2.06/0.73 = { by lemma 3 R->L }
% 2.06/0.73 multiply(multiply(inverse(multiply(inverse(T2), multiply(T2, inverse(multiply(multiply(inverse(multiply(inverse(V), X)), U), inverse(multiply(V, U))))))), W), inverse(multiply(Z, W)))
% 2.06/0.73 = { by lemma 6 }
% 2.06/0.73 multiply(multiply(inverse(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(V), X)), U), inverse(multiply(V, U))))))), W), inverse(multiply(Z, W)))
% 2.06/0.73 = { by lemma 5 R->L }
% 2.06/0.74 multiply(inverse(T), multiply(T, multiply(S, inverse(multiply(multiply(inverse(multiply(multiply(inverse(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(V), X)), U), inverse(multiply(V, U))))))), W), inverse(multiply(Z, W)))), X2), inverse(multiply(inverse(S), X2)))))))
% 2.06/0.74 = { by axiom 1 (single_axiom) R->L }
% 2.06/0.74 multiply(inverse(T), multiply(T, multiply(S, inverse(multiply(multiply(inverse(multiply(multiply(inverse(multiply(inverse(Z), multiply(inverse(inverse(Y2)), inverse(multiply(multiply(inverse(multiply(inverse(Z2), multiply(inverse(inverse(inverse(Y2))), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(V), X)), U), inverse(multiply(V, U)))))))), W2), inverse(multiply(Z2, W2))))))), W), inverse(multiply(Z, W)))), X2), inverse(multiply(inverse(S), X2)))))))
% 2.06/0.74 = { by lemma 2 }
% 2.06/0.74 multiply(inverse(T), multiply(T, multiply(S, inverse(multiply(multiply(multiply(Y2, inverse(multiply(multiply(inverse(multiply(inverse(V2), inverse(multiply(multiply(inverse(multiply(inverse(Z2), multiply(inverse(inverse(inverse(Y2))), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(V), X)), U), inverse(multiply(V, U)))))))), W2), inverse(multiply(Z2, W2)))))), U2), inverse(multiply(V2, U2))))), X2), inverse(multiply(inverse(S), X2)))))))
% 2.06/0.74 = { by lemma 2 R->L }
% 2.06/0.74 multiply(inverse(T), multiply(T, multiply(S, inverse(multiply(multiply(inverse(multiply(multiply(inverse(multiply(inverse(Z), multiply(inverse(inverse(Y2)), inverse(multiply(multiply(inverse(multiply(inverse(Z2), multiply(inverse(inverse(inverse(Y2))), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(V), X)), U), inverse(multiply(V, U)))))))), W2), inverse(multiply(Z2, W2))))))), Y), inverse(multiply(Z, Y)))), X2), inverse(multiply(inverse(S), X2)))))))
% 2.06/0.74 = { by axiom 1 (single_axiom) }
% 2.06/0.74 multiply(inverse(T), multiply(T, multiply(S, inverse(multiply(multiply(inverse(multiply(multiply(inverse(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(V), X)), U), inverse(multiply(V, U))))))), Y), inverse(multiply(Z, Y)))), X2), inverse(multiply(inverse(S), X2)))))))
% 2.06/0.74 = { by lemma 5 }
% 2.06/0.74 multiply(multiply(inverse(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(V), X)), U), inverse(multiply(V, U))))))), Y), inverse(multiply(Z, Y)))
% 2.06/0.74 = { by lemma 3 }
% 2.06/0.74 multiply(multiply(inverse(X), Y), inverse(multiply(Z, Y)))
% 2.06/0.74
% 2.06/0.74 Lemma 8: multiply(X, multiply(Y, inverse(multiply(multiply(inverse(multiply(inverse(X), Z)), W), inverse(multiply(inverse(Y), W)))))) = Z.
% 2.06/0.74 Proof:
% 2.06/0.74 multiply(X, multiply(Y, inverse(multiply(multiply(inverse(multiply(inverse(X), Z)), W), inverse(multiply(inverse(Y), W))))))
% 2.06/0.74 = { by lemma 4 R->L }
% 2.06/0.74 multiply(X, inverse(multiply(multiply(inverse(multiply(inverse(V), multiply(inverse(X), Z))), U), inverse(multiply(V, U)))))
% 2.06/0.74 = { by axiom 1 (single_axiom) }
% 2.06/0.74 Z
% 2.06/0.74
% 2.06/0.74 Lemma 9: multiply(Y, inverse(Y)) = multiply(X, inverse(X)).
% 2.06/0.74 Proof:
% 2.06/0.74 multiply(Y, inverse(Y))
% 2.06/0.74 = { by lemma 8 R->L }
% 2.06/0.74 multiply(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(inverse(Z)), Y)), V), inverse(multiply(inverse(Z), V)))))), inverse(Y))
% 2.06/0.74 = { by lemma 3 R->L }
% 2.06/0.74 multiply(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(inverse(Z)), Y)), V), inverse(multiply(inverse(Z), V)))))), inverse(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(inverse(Z)), Y)), V), inverse(multiply(inverse(Z), V))))))))
% 2.06/0.74 = { by lemma 7 R->L }
% 2.06/0.74 multiply(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(inverse(Z)), X)), W), inverse(multiply(inverse(Z), W)))))), inverse(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(inverse(Z)), X)), W), inverse(multiply(inverse(Z), W))))))))
% 2.06/0.74 = { by lemma 3 }
% 2.06/0.74 multiply(multiply(inverse(Z), multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(inverse(Z)), X)), W), inverse(multiply(inverse(Z), W)))))), inverse(X))
% 2.06/0.74 = { by lemma 8 }
% 2.06/0.74 multiply(X, inverse(X))
% 2.06/0.74
% 2.06/0.74 Lemma 10: multiply(multiply(inverse(X), Y), inverse(multiply(Z, Y))) = multiply(multiply(W, inverse(W)), inverse(multiply(Z, inverse(inverse(X))))).
% 2.06/0.74 Proof:
% 2.06/0.74 multiply(multiply(inverse(X), Y), inverse(multiply(Z, Y)))
% 2.06/0.74 = { by lemma 7 }
% 2.06/0.74 multiply(multiply(inverse(X), inverse(inverse(X))), inverse(multiply(Z, inverse(inverse(X)))))
% 2.06/0.74 = { by lemma 9 R->L }
% 2.06/0.74 multiply(multiply(W, inverse(W)), inverse(multiply(Z, inverse(inverse(X)))))
% 2.06/0.74
% 2.06/0.74 Lemma 11: multiply(X, inverse(multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(X), inverse(inverse(Z))))))) = multiply(Z, inverse(multiply(W, inverse(W)))).
% 2.06/0.74 Proof:
% 2.06/0.74 multiply(X, inverse(multiply(multiply(Y, inverse(Y)), inverse(multiply(inverse(X), inverse(inverse(Z)))))))
% 2.06/0.74 = { by lemma 10 R->L }
% 2.06/0.74 multiply(X, inverse(multiply(multiply(inverse(Z), V), inverse(multiply(inverse(X), V)))))
% 2.06/0.75 = { by lemma 3 R->L }
% 2.06/0.75 multiply(X, inverse(multiply(multiply(inverse(multiply(inverse(inverse(X)), multiply(inverse(X), inverse(multiply(multiply(inverse(multiply(inverse(U), Z)), T), inverse(multiply(U, T))))))), V), inverse(multiply(inverse(X), V)))))
% 2.06/0.75 = { by axiom 1 (single_axiom) }
% 2.06/0.75 inverse(multiply(multiply(inverse(multiply(inverse(U), Z)), T), inverse(multiply(U, T))))
% 2.06/0.75 = { by axiom 1 (single_axiom) R->L }
% 2.06/0.75 multiply(Z, inverse(multiply(multiply(inverse(multiply(inverse(inverse(Z)), multiply(inverse(Z), inverse(multiply(multiply(inverse(multiply(inverse(U), Z)), T), inverse(multiply(U, T))))))), S), inverse(multiply(inverse(Z), S)))))
% 2.06/0.75 = { by lemma 3 }
% 2.06/0.75 multiply(Z, inverse(multiply(multiply(inverse(Z), S), inverse(multiply(inverse(Z), S)))))
% 2.06/0.75 = { by lemma 9 R->L }
% 2.06/0.75 multiply(Z, inverse(multiply(W, inverse(W))))
% 2.06/0.76
% 2.06/0.76 Lemma 12: multiply(multiply(inverse(X), multiply(X, Y)), inverse(multiply(Z, inverse(Z)))) = Y.
% 2.06/0.76 Proof:
% 2.06/0.76 multiply(multiply(inverse(X), multiply(X, Y)), inverse(multiply(Z, inverse(Z))))
% 2.06/0.76 = { by lemma 11 R->L }
% 2.06/0.76 multiply(W, inverse(multiply(multiply(inverse(multiply(inverse(X), multiply(X, Y))), inverse(inverse(multiply(inverse(X), multiply(X, Y))))), inverse(multiply(inverse(W), inverse(inverse(multiply(inverse(X), multiply(X, Y)))))))))
% 2.06/0.76 = { by lemma 6 }
% 2.06/0.76 multiply(W, inverse(multiply(multiply(inverse(multiply(inverse(inverse(W)), multiply(inverse(W), Y))), inverse(inverse(multiply(inverse(X), multiply(X, Y))))), inverse(multiply(inverse(W), inverse(inverse(multiply(inverse(X), multiply(X, Y)))))))))
% 2.06/0.76 = { by axiom 1 (single_axiom) }
% 2.06/0.76 Y
% 2.06/0.76
% 2.06/0.76 Lemma 13: multiply(inverse(inverse(X)), multiply(inverse(Y), multiply(Y, Z))) = multiply(inverse(W), multiply(W, multiply(X, Z))).
% 2.06/0.76 Proof:
% 2.06/0.76 multiply(inverse(inverse(X)), multiply(inverse(Y), multiply(Y, Z)))
% 2.06/0.76 = { by lemma 6 }
% 2.06/0.76 multiply(inverse(inverse(X)), multiply(inverse(X), multiply(X, Z)))
% 2.06/0.76 = { by lemma 6 R->L }
% 2.06/0.76 multiply(inverse(W), multiply(W, multiply(X, Z)))
% 2.06/0.76
% 2.06/0.76 Lemma 14: multiply(inverse(Z), multiply(Z, inverse(X))) = multiply(inverse(X), multiply(Y, inverse(Y))).
% 2.06/0.76 Proof:
% 2.06/0.76 multiply(inverse(Z), multiply(Z, inverse(X)))
% 2.06/0.76 = { by lemma 6 }
% 2.06/0.76 multiply(inverse(X), multiply(X, inverse(X)))
% 2.06/0.76 = { by lemma 9 R->L }
% 2.06/0.76 multiply(inverse(X), multiply(Y, inverse(Y)))
% 2.06/0.76
% 2.06/0.76 Lemma 15: multiply(inverse(multiply(inverse(X), multiply(X, Y))), Y) = multiply(inverse(Z), multiply(Z, inverse(multiply(W, inverse(W))))).
% 2.06/0.76 Proof:
% 2.06/0.76 multiply(inverse(multiply(inverse(X), multiply(X, Y))), Y)
% 2.06/0.76 = { by lemma 12 R->L }
% 2.06/0.76 multiply(inverse(multiply(inverse(X), multiply(X, Y))), multiply(multiply(inverse(X), multiply(X, Y)), inverse(multiply(W, inverse(W)))))
% 2.06/0.76 = { by lemma 6 R->L }
% 2.06/0.76 multiply(inverse(Z), multiply(Z, inverse(multiply(W, inverse(W)))))
% 2.06/0.76
% 2.06/0.76 Lemma 16: inverse(multiply(inverse(X), multiply(X, multiply(Y, inverse(Y))))) = inverse(multiply(Z, inverse(Z))).
% 2.06/0.76 Proof:
% 2.06/0.76 inverse(multiply(inverse(X), multiply(X, multiply(Y, inverse(Y)))))
% 2.06/0.76 = { by lemma 12 R->L }
% 2.06/0.76 multiply(multiply(inverse(W), multiply(W, inverse(multiply(inverse(X), multiply(X, multiply(Y, inverse(Y))))))), inverse(multiply(V, inverse(V))))
% 2.06/0.76 = { by lemma 14 }
% 2.06/0.76 multiply(multiply(inverse(multiply(inverse(X), multiply(X, multiply(Y, inverse(Y))))), multiply(Y, inverse(Y))), inverse(multiply(V, inverse(V))))
% 2.06/0.76 = { by lemma 15 }
% 2.06/0.76 multiply(multiply(inverse(U), multiply(U, inverse(multiply(Z, inverse(Z))))), inverse(multiply(V, inverse(V))))
% 2.06/0.76 = { by lemma 12 }
% 2.06/0.76 inverse(multiply(Z, inverse(Z)))
% 2.06/0.76
% 2.06/0.76 Lemma 17: multiply(X, multiply(Y, inverse(Y))) = inverse(inverse(X)).
% 2.06/0.76 Proof:
% 2.06/0.76 multiply(X, multiply(Y, inverse(Y)))
% 2.06/0.76 = { by lemma 12 R->L }
% 2.06/0.76 multiply(multiply(inverse(Z), multiply(Z, multiply(X, multiply(Y, inverse(Y))))), inverse(multiply(W, inverse(W))))
% 2.06/0.76 = { by lemma 13 R->L }
% 2.06/0.76 multiply(multiply(inverse(inverse(X)), multiply(inverse(V), multiply(V, multiply(Y, inverse(Y))))), inverse(multiply(W, inverse(W))))
% 2.06/0.76 = { by lemma 5 R->L }
% 2.06/0.76 multiply(multiply(inverse(inverse(X)), multiply(inverse(U), multiply(U, multiply(T, inverse(multiply(multiply(inverse(multiply(inverse(V), multiply(V, multiply(Y, inverse(Y))))), S), inverse(multiply(inverse(T), S)))))))), inverse(multiply(W, inverse(W))))
% 2.06/0.76 = { by lemma 16 }
% 2.06/0.76 multiply(multiply(inverse(inverse(X)), multiply(inverse(U), multiply(U, multiply(T, inverse(multiply(multiply(inverse(multiply(inverse(X), inverse(inverse(X)))), S), inverse(multiply(inverse(T), S)))))))), inverse(multiply(W, inverse(W))))
% 2.06/0.76 = { by lemma 5 }
% 2.06/0.76 multiply(multiply(inverse(inverse(X)), multiply(inverse(X), inverse(inverse(X)))), inverse(multiply(W, inverse(W))))
% 2.06/0.76 = { by lemma 12 }
% 2.06/0.76 inverse(inverse(X))
% 2.06/0.76
% 2.06/0.76 Lemma 18: multiply(inverse(X), multiply(X, inverse(Y))) = inverse(inverse(inverse(Y))).
% 2.06/0.76 Proof:
% 2.06/0.76 multiply(inverse(X), multiply(X, inverse(Y)))
% 2.06/0.76 = { by lemma 14 }
% 2.06/0.76 multiply(inverse(Y), multiply(Z, inverse(Z)))
% 2.06/0.76 = { by lemma 17 }
% 2.06/0.76 inverse(inverse(inverse(Y)))
% 2.06/0.76
% 2.06/0.76 Lemma 19: inverse(inverse(inverse(multiply(X, inverse(X))))) = multiply(inverse(inverse(inverse(inverse(Y)))), inverse(Y)).
% 2.06/0.76 Proof:
% 2.06/0.76 inverse(inverse(inverse(multiply(X, inverse(X)))))
% 2.06/0.76 = { by lemma 18 R->L }
% 2.06/0.76 multiply(inverse(Z), multiply(Z, inverse(multiply(X, inverse(X)))))
% 2.06/0.76 = { by lemma 15 R->L }
% 2.06/0.76 multiply(inverse(multiply(inverse(W), multiply(W, inverse(Y)))), inverse(Y))
% 2.06/0.76 = { by lemma 18 }
% 2.06/0.76 multiply(inverse(inverse(inverse(inverse(Y)))), inverse(Y))
% 2.06/0.76
% 2.06/0.76 Lemma 20: inverse(multiply(X, inverse(X))) = multiply(inverse(inverse(inverse(inverse(Y)))), inverse(Y)).
% 2.06/0.76 Proof:
% 2.06/0.76 inverse(multiply(X, inverse(X)))
% 2.06/0.76 = { by axiom 1 (single_axiom) R->L }
% 2.06/0.76 multiply(inverse(inverse(Z)), inverse(multiply(multiply(inverse(multiply(inverse(W), multiply(inverse(inverse(inverse(Z))), inverse(multiply(X, inverse(X)))))), V), inverse(multiply(W, V)))))
% 2.06/0.76 = { by lemma 17 R->L }
% 2.06/0.76 multiply(inverse(inverse(Z)), inverse(multiply(multiply(inverse(multiply(inverse(W), multiply(multiply(inverse(Z), multiply(Z, inverse(Z))), inverse(multiply(X, inverse(X)))))), V), inverse(multiply(W, V)))))
% 2.06/0.76 = { by lemma 12 }
% 2.06/0.76 multiply(inverse(inverse(Z)), inverse(multiply(multiply(inverse(multiply(inverse(W), inverse(Z))), V), inverse(multiply(W, V)))))
% 2.06/0.76 = { by lemma 4 }
% 2.06/0.76 multiply(inverse(inverse(Z)), multiply(inverse(Z), inverse(multiply(multiply(inverse(inverse(Z)), U), inverse(multiply(inverse(inverse(Z)), U))))))
% 2.06/0.76 = { by lemma 18 }
% 2.06/0.76 inverse(inverse(inverse(multiply(multiply(inverse(inverse(Z)), U), inverse(multiply(inverse(inverse(Z)), U))))))
% 2.06/0.76 = { by lemma 19 }
% 2.06/0.76 multiply(inverse(inverse(inverse(inverse(Y)))), inverse(Y))
% 2.06/0.76
% 2.06/0.76 Lemma 21: inverse(inverse(multiply(X, multiply(inverse(inverse(inverse(inverse(Y)))), inverse(Y))))) = X.
% 2.06/0.76 Proof:
% 2.06/0.76 inverse(inverse(multiply(X, multiply(inverse(inverse(inverse(inverse(Y)))), inverse(Y)))))
% 2.06/0.76 = { by lemma 20 R->L }
% 2.06/0.76 inverse(inverse(multiply(X, inverse(multiply(Z, inverse(Z))))))
% 2.06/0.76 = { by lemma 11 R->L }
% 2.06/0.76 inverse(inverse(multiply(W, inverse(multiply(multiply(inverse(X), inverse(inverse(X))), inverse(multiply(inverse(W), inverse(inverse(X)))))))))
% 2.06/0.76 = { by lemma 4 R->L }
% 2.06/0.76 inverse(inverse(inverse(multiply(multiply(inverse(multiply(inverse(inverse(V)), X)), U), inverse(multiply(inverse(V), U))))))
% 2.06/0.76 = { by lemma 18 R->L }
% 2.06/0.76 multiply(inverse(V), multiply(V, inverse(multiply(multiply(inverse(multiply(inverse(inverse(V)), X)), U), inverse(multiply(inverse(V), U))))))
% 2.06/0.76 = { by lemma 8 }
% 2.06/0.76 X
% 2.06/0.76
% 2.06/0.76 Lemma 22: multiply(inverse(inverse(inverse(inverse(X)))), inverse(X)) = multiply(inverse(inverse(inverse(Y))), Y).
% 2.06/0.76 Proof:
% 2.06/0.76 multiply(inverse(inverse(inverse(inverse(X)))), inverse(X))
% 2.06/0.76 = { by lemma 18 R->L }
% 2.06/0.76 multiply(inverse(multiply(inverse(Z), multiply(Z, inverse(X)))), inverse(X))
% 2.06/0.76 = { by lemma 15 }
% 2.06/0.76 multiply(inverse(W), multiply(W, inverse(multiply(V, inverse(V)))))
% 2.06/0.76 = { by lemma 15 R->L }
% 2.06/0.76 multiply(inverse(multiply(inverse(U), multiply(U, inverse(inverse(multiply(Y, multiply(inverse(inverse(inverse(inverse(T)))), inverse(T)))))))), inverse(inverse(multiply(Y, multiply(inverse(inverse(inverse(inverse(T)))), inverse(T))))))
% 2.06/0.76 = { by lemma 18 }
% 2.06/0.76 multiply(inverse(inverse(inverse(inverse(inverse(multiply(Y, multiply(inverse(inverse(inverse(inverse(T)))), inverse(T)))))))), inverse(inverse(multiply(Y, multiply(inverse(inverse(inverse(inverse(T)))), inverse(T))))))
% 2.06/0.76 = { by lemma 21 }
% 2.06/0.76 multiply(inverse(inverse(inverse(Y))), inverse(inverse(multiply(Y, multiply(inverse(inverse(inverse(inverse(T)))), inverse(T))))))
% 2.06/0.76 = { by lemma 21 }
% 2.06/0.76 multiply(inverse(inverse(inverse(Y))), Y)
% 2.06/0.76
% 2.06/0.76 Lemma 23: multiply(inverse(X), multiply(X, Y)) = inverse(inverse(Y)).
% 2.06/0.76 Proof:
% 2.06/0.76 multiply(inverse(X), multiply(X, Y))
% 2.06/0.76 = { by lemma 21 R->L }
% 2.06/0.76 multiply(inverse(X), multiply(X, inverse(inverse(multiply(Y, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)))))))
% 2.06/0.76 = { by lemma 18 }
% 2.06/0.76 inverse(inverse(inverse(inverse(multiply(Y, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)))))))
% 2.06/0.76 = { by lemma 21 }
% 2.06/0.76 inverse(inverse(Y))
% 2.06/0.76
% 2.06/0.76 Lemma 24: inverse(inverse(multiply(X, Y))) = multiply(inverse(inverse(X)), inverse(inverse(Y))).
% 2.06/0.76 Proof:
% 2.06/0.76 inverse(inverse(multiply(X, Y)))
% 2.06/0.76 = { by lemma 23 R->L }
% 2.06/0.76 multiply(inverse(Z), multiply(Z, multiply(X, Y)))
% 2.06/0.76 = { by lemma 13 R->L }
% 2.06/0.76 multiply(inverse(inverse(X)), multiply(inverse(W), multiply(W, Y)))
% 2.06/0.76 = { by lemma 23 }
% 2.06/0.77 multiply(inverse(inverse(X)), inverse(inverse(Y)))
% 2.06/0.77
% 2.06/0.77 Lemma 25: multiply(inverse(inverse(inverse(X))), X) = multiply(Y, inverse(Y)).
% 2.06/0.77 Proof:
% 2.06/0.77 multiply(inverse(inverse(inverse(X))), X)
% 2.06/0.77 = { by lemma 22 R->L }
% 2.06/0.77 multiply(inverse(inverse(inverse(inverse(inverse(multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))))), inverse(inverse(multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)))))
% 2.06/0.77 = { by lemma 24 R->L }
% 2.06/0.77 inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))), multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)))))
% 2.06/0.77 = { by lemma 21 }
% 2.06/0.77 inverse(inverse(inverse(multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)))))
% 2.06/0.77 = { by lemma 24 }
% 2.06/0.77 inverse(multiply(inverse(inverse(inverse(inverse(inverse(inverse(Z)))))), inverse(inverse(inverse(Z)))))
% 2.06/0.77 = { by lemma 19 R->L }
% 2.06/0.77 inverse(inverse(inverse(inverse(multiply(W, inverse(W))))))
% 2.06/0.77 = { by lemma 18 R->L }
% 2.06/0.77 multiply(inverse(inverse(inverse(V))), multiply(inverse(inverse(V)), inverse(inverse(multiply(W, inverse(W))))))
% 2.06/0.77 = { by lemma 24 R->L }
% 2.06/0.77 multiply(inverse(inverse(inverse(V))), inverse(inverse(multiply(V, multiply(W, inverse(W))))))
% 2.06/0.77 = { by lemma 24 R->L }
% 2.06/0.77 inverse(inverse(multiply(inverse(V), multiply(V, multiply(W, inverse(W))))))
% 2.06/0.77 = { by lemma 16 }
% 2.06/0.77 inverse(inverse(multiply(U, inverse(U))))
% 2.06/0.77 = { by lemma 24 }
% 2.06/0.77 multiply(inverse(inverse(U)), inverse(inverse(inverse(U))))
% 2.06/0.77 = { by lemma 9 R->L }
% 2.06/0.77 multiply(Y, inverse(Y))
% 2.06/0.77
% 2.06/0.77 Lemma 26: multiply(inverse(inverse(X)), multiply(inverse(inverse(inverse(Y))), Y)) = X.
% 2.06/0.77 Proof:
% 2.06/0.77 multiply(inverse(inverse(X)), multiply(inverse(inverse(inverse(Y))), Y))
% 2.06/0.77 = { by lemma 22 R->L }
% 2.06/0.77 multiply(inverse(inverse(X)), multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)))
% 2.06/0.77 = { by lemma 20 R->L }
% 2.06/0.77 multiply(inverse(inverse(X)), inverse(multiply(multiply(inverse(inverse(inverse(X))), W), inverse(multiply(inverse(inverse(inverse(X))), W)))))
% 2.06/0.77 = { by lemma 4 R->L }
% 2.06/0.77 inverse(multiply(multiply(inverse(multiply(inverse(inverse(inverse(inverse(V)))), inverse(inverse(X)))), inverse(inverse(U))), inverse(multiply(inverse(inverse(inverse(V))), inverse(inverse(U))))))
% 2.06/0.77 = { by lemma 24 R->L }
% 2.06/0.77 inverse(multiply(multiply(inverse(multiply(inverse(inverse(inverse(inverse(V)))), inverse(inverse(X)))), inverse(inverse(U))), inverse(inverse(inverse(multiply(inverse(V), U))))))
% 2.06/0.77 = { by lemma 24 R->L }
% 2.06/0.77 inverse(multiply(multiply(inverse(inverse(inverse(multiply(inverse(inverse(V)), X)))), inverse(inverse(U))), inverse(inverse(inverse(multiply(inverse(V), U))))))
% 2.06/0.77 = { by lemma 24 R->L }
% 2.06/0.77 inverse(multiply(inverse(inverse(multiply(inverse(multiply(inverse(inverse(V)), X)), U))), inverse(inverse(inverse(multiply(inverse(V), U))))))
% 2.06/0.77 = { by lemma 24 R->L }
% 2.06/0.77 inverse(inverse(inverse(multiply(multiply(inverse(multiply(inverse(inverse(V)), X)), U), inverse(multiply(inverse(V), U))))))
% 2.06/0.77 = { by lemma 18 R->L }
% 2.06/0.77 multiply(inverse(V), multiply(V, inverse(multiply(multiply(inverse(multiply(inverse(inverse(V)), X)), U), inverse(multiply(inverse(V), U))))))
% 2.06/0.77 = { by lemma 8 }
% 2.06/0.77 X
% 2.06/0.77
% 2.06/0.77 Lemma 27: multiply(multiply(X, multiply(Y, inverse(multiply(Z, inverse(Z))))), inverse(multiply(W, inverse(W)))) = inverse(multiply(multiply(V, inverse(V)), inverse(multiply(X, inverse(inverse(Y)))))).
% 2.06/0.77 Proof:
% 2.06/0.77 multiply(multiply(X, multiply(Y, inverse(multiply(Z, inverse(Z))))), inverse(multiply(W, inverse(W))))
% 2.06/0.77 = { by lemma 11 R->L }
% 2.06/0.77 multiply(multiply(X, multiply(U, inverse(multiply(multiply(inverse(Y), inverse(inverse(Y))), inverse(multiply(inverse(U), inverse(inverse(Y)))))))), inverse(multiply(W, inverse(W))))
% 2.06/0.77 = { by lemma 4 R->L }
% 2.06/0.77 multiply(multiply(X, inverse(multiply(multiply(inverse(multiply(inverse(inverse(X)), Y)), T), inverse(multiply(inverse(X), T))))), inverse(multiply(W, inverse(W))))
% 2.06/0.77 = { by lemma 4 R->L }
% 2.06/0.77 multiply(inverse(multiply(multiply(inverse(multiply(inverse(S), multiply(inverse(inverse(X)), Y))), X2), inverse(multiply(S, X2)))), inverse(multiply(W, inverse(W))))
% 2.06/0.77 = { by lemma 3 R->L }
% 2.06/0.77 multiply(multiply(inverse(Y2), multiply(Y2, inverse(multiply(multiply(inverse(multiply(inverse(X), inverse(multiply(multiply(inverse(multiply(inverse(S), multiply(inverse(inverse(X)), Y))), X2), inverse(multiply(S, X2)))))), Z2), inverse(multiply(X, Z2)))))), inverse(multiply(W, inverse(W))))
% 2.06/0.77 = { by axiom 1 (single_axiom) }
% 2.06/0.77 multiply(multiply(inverse(Y2), multiply(Y2, inverse(multiply(multiply(inverse(Y), Z2), inverse(multiply(X, Z2)))))), inverse(multiply(W, inverse(W))))
% 2.06/0.77 = { by lemma 12 }
% 2.06/0.77 inverse(multiply(multiply(inverse(Y), Z2), inverse(multiply(X, Z2))))
% 2.06/0.77 = { by lemma 10 }
% 2.06/0.77 inverse(multiply(multiply(V, inverse(V)), inverse(multiply(X, inverse(inverse(Y))))))
% 2.06/0.77
% 2.06/0.77 Lemma 28: multiply(multiply(X, inverse(X)), inverse(inverse(inverse(Y)))) = inverse(inverse(inverse(Y))).
% 2.06/0.77 Proof:
% 2.06/0.77 multiply(multiply(X, inverse(X)), inverse(inverse(inverse(Y))))
% 2.06/0.77 = { by lemma 17 R->L }
% 2.06/0.77 multiply(multiply(X, inverse(X)), inverse(multiply(Y, multiply(inverse(inverse(Z)), inverse(inverse(inverse(Z)))))))
% 2.06/0.77 = { by lemma 24 R->L }
% 2.06/0.77 multiply(multiply(X, inverse(X)), inverse(multiply(Y, inverse(inverse(multiply(Z, inverse(Z)))))))
% 2.06/0.77 = { by lemma 26 R->L }
% 2.06/0.78 multiply(inverse(inverse(multiply(multiply(X, inverse(X)), inverse(multiply(Y, inverse(inverse(multiply(Z, inverse(Z))))))))), multiply(inverse(inverse(inverse(W))), W))
% 2.06/0.78 = { by lemma 27 R->L }
% 2.06/0.78 multiply(inverse(multiply(multiply(Y, multiply(multiply(Z, inverse(Z)), inverse(multiply(Z, inverse(Z))))), inverse(multiply(V, inverse(V))))), multiply(inverse(inverse(inverse(W))), W))
% 2.06/0.78 = { by lemma 17 }
% 2.06/0.78 multiply(inverse(multiply(inverse(inverse(Y)), inverse(multiply(V, inverse(V))))), multiply(inverse(inverse(inverse(W))), W))
% 2.06/0.78 = { by lemma 20 }
% 2.06/0.78 multiply(inverse(multiply(inverse(inverse(Y)), multiply(inverse(inverse(inverse(inverse(U)))), inverse(U)))), multiply(inverse(inverse(inverse(W))), W))
% 2.06/0.78 = { by lemma 26 }
% 2.06/0.78 multiply(inverse(Y), multiply(inverse(inverse(inverse(W))), W))
% 2.06/0.78 = { by lemma 25 }
% 2.06/0.78 multiply(inverse(Y), multiply(Y, inverse(Y)))
% 2.06/0.78 = { by lemma 18 }
% 2.06/0.78 inverse(inverse(inverse(Y)))
% 2.06/0.78
% 2.06/0.78 Lemma 29: inverse(inverse(inverse(inverse(X)))) = X.
% 2.06/0.78 Proof:
% 2.06/0.78 inverse(inverse(inverse(inverse(X))))
% 2.06/0.78 = { by lemma 18 R->L }
% 2.06/0.78 multiply(inverse(inverse(X)), multiply(inverse(X), inverse(inverse(X))))
% 2.06/0.78 = { by lemma 25 R->L }
% 2.06/0.78 multiply(inverse(inverse(X)), multiply(inverse(inverse(inverse(Y))), Y))
% 2.06/0.78 = { by lemma 26 }
% 2.06/0.78 X
% 2.06/0.78
% 2.06/0.78 Lemma 30: inverse(inverse(inverse(X))) = inverse(X).
% 2.06/0.78 Proof:
% 2.06/0.78 inverse(inverse(inverse(X)))
% 2.06/0.78 = { by lemma 28 R->L }
% 2.06/0.78 multiply(multiply(Y, inverse(Y)), inverse(inverse(inverse(X))))
% 2.06/0.78 = { by lemma 25 R->L }
% 2.06/0.78 multiply(multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)), inverse(inverse(inverse(X))))
% 2.06/0.78 = { by lemma 20 R->L }
% 2.06/0.78 multiply(inverse(multiply(W, inverse(W))), inverse(inverse(inverse(X))))
% 2.06/0.78 = { by lemma 28 R->L }
% 2.06/0.78 multiply(inverse(multiply(W, inverse(W))), multiply(multiply(W, inverse(W)), inverse(inverse(inverse(X)))))
% 2.06/0.78 = { by lemma 18 }
% 2.06/0.78 inverse(inverse(inverse(inverse(inverse(X)))))
% 2.06/0.78 = { by lemma 29 }
% 2.06/0.78 inverse(X)
% 2.06/0.78
% 2.06/0.78 Lemma 31: inverse(inverse(X)) = X.
% 2.06/0.78 Proof:
% 2.06/0.78 inverse(inverse(X))
% 2.06/0.78 = { by lemma 17 R->L }
% 2.06/0.78 multiply(X, multiply(Y, inverse(Y)))
% 2.06/0.78 = { by lemma 25 R->L }
% 2.06/0.78 multiply(X, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)))
% 2.06/0.78 = { by lemma 20 R->L }
% 2.06/0.78 multiply(X, inverse(multiply(multiply(inverse(X), inverse(inverse(inverse(inverse(X))))), inverse(multiply(inverse(X), inverse(inverse(inverse(inverse(X)))))))))
% 2.06/0.78 = { by lemma 30 R->L }
% 2.06/0.78 multiply(X, inverse(multiply(multiply(inverse(inverse(inverse(X))), inverse(inverse(inverse(inverse(X))))), inverse(multiply(inverse(X), inverse(inverse(inverse(inverse(X)))))))))
% 2.06/0.78 = { by lemma 11 }
% 2.06/0.78 multiply(inverse(inverse(X)), inverse(multiply(W, inverse(W))))
% 2.06/0.78 = { by lemma 20 }
% 2.06/0.78 multiply(inverse(inverse(X)), multiply(inverse(inverse(inverse(inverse(V)))), inverse(V)))
% 2.06/0.78 = { by lemma 26 }
% 2.06/0.78 X
% 2.06/0.78
% 2.06/0.78 Lemma 32: multiply(inverse(X), X) = multiply(Y, inverse(Y)).
% 2.06/0.78 Proof:
% 2.06/0.78 multiply(inverse(X), X)
% 2.06/0.78 = { by lemma 30 R->L }
% 2.06/0.78 multiply(inverse(inverse(inverse(X))), X)
% 2.06/0.78 = { by lemma 25 }
% 2.06/0.78 multiply(Y, inverse(Y))
% 2.06/0.78
% 2.06/0.78 Lemma 33: multiply(X, inverse(multiply(inverse(Y), X))) = Y.
% 2.06/0.78 Proof:
% 2.06/0.78 multiply(X, inverse(multiply(inverse(Y), X)))
% 2.06/0.78 = { by lemma 31 R->L }
% 2.06/0.78 multiply(X, inverse(multiply(inverse(Y), inverse(inverse(X)))))
% 2.06/0.78 = { by lemma 29 R->L }
% 2.06/0.78 multiply(X, inverse(multiply(inverse(inverse(inverse(inverse(inverse(Y))))), inverse(inverse(X)))))
% 2.06/0.78 = { by lemma 24 R->L }
% 2.06/0.78 multiply(X, inverse(inverse(inverse(multiply(inverse(inverse(inverse(Y))), X)))))
% 2.06/0.78 = { by lemma 17 R->L }
% 2.06/0.78 multiply(X, inverse(multiply(multiply(inverse(inverse(inverse(Y))), X), multiply(Z, inverse(Z)))))
% 2.06/0.78 = { by lemma 25 R->L }
% 2.06/0.78 multiply(X, inverse(multiply(multiply(inverse(inverse(inverse(Y))), X), multiply(inverse(inverse(inverse(inverse(W)))), inverse(W)))))
% 2.06/0.78 = { by lemma 19 R->L }
% 2.06/0.78 multiply(X, inverse(multiply(multiply(inverse(inverse(inverse(Y))), X), inverse(inverse(inverse(multiply(V, inverse(V))))))))
% 2.06/0.78 = { by lemma 31 R->L }
% 2.06/0.78 multiply(X, inverse(multiply(multiply(inverse(inverse(inverse(Y))), inverse(inverse(X))), inverse(inverse(inverse(multiply(V, inverse(V))))))))
% 2.06/0.78 = { by lemma 24 R->L }
% 2.06/0.78 multiply(X, inverse(multiply(inverse(inverse(multiply(inverse(Y), X))), inverse(inverse(inverse(multiply(V, inverse(V))))))))
% 2.06/0.78 = { by lemma 24 R->L }
% 2.06/0.78 multiply(X, inverse(inverse(inverse(multiply(multiply(inverse(Y), X), inverse(multiply(V, inverse(V))))))))
% 2.06/0.78 = { by lemma 31 R->L }
% 2.06/0.78 multiply(inverse(inverse(X)), inverse(inverse(inverse(multiply(multiply(inverse(Y), X), inverse(multiply(V, inverse(V))))))))
% 2.06/0.78 = { by lemma 24 R->L }
% 2.06/0.78 inverse(inverse(multiply(X, inverse(multiply(multiply(inverse(Y), X), inverse(multiply(V, inverse(V))))))))
% 2.06/0.78 = { by lemma 23 R->L }
% 2.06/0.78 multiply(inverse(U), multiply(U, multiply(X, inverse(multiply(multiply(inverse(Y), X), inverse(multiply(V, inverse(V))))))))
% 2.06/0.78 = { by lemma 32 R->L }
% 2.06/0.78 multiply(inverse(U), multiply(U, multiply(X, inverse(multiply(multiply(inverse(Y), X), inverse(multiply(inverse(X), X)))))))
% 2.06/0.78 = { by lemma 5 }
% 2.06/0.79 Y
% 2.06/0.79
% 2.06/0.79 Lemma 34: multiply(multiply(X, inverse(X)), Y) = inverse(inverse(Y)).
% 2.06/0.79 Proof:
% 2.06/0.79 multiply(multiply(X, inverse(X)), Y)
% 2.06/0.79 = { by lemma 25 R->L }
% 2.06/0.79 multiply(multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)), Y)
% 2.06/0.79 = { by lemma 20 R->L }
% 2.06/0.79 multiply(inverse(multiply(W, inverse(W))), Y)
% 2.06/0.79 = { by lemma 33 R->L }
% 2.06/0.79 multiply(multiply(W, inverse(W)), inverse(multiply(inverse(multiply(inverse(multiply(W, inverse(W))), Y)), multiply(W, inverse(W)))))
% 2.06/0.79 = { by lemma 5 R->L }
% 2.06/0.79 multiply(multiply(W, inverse(W)), inverse(multiply(inverse(multiply(inverse(multiply(W, inverse(W))), Y)), multiply(inverse(V), multiply(V, multiply(U, inverse(multiply(multiply(inverse(multiply(W, inverse(W))), T), inverse(multiply(inverse(U), T))))))))))
% 2.06/0.79 = { by lemma 12 R->L }
% 2.06/0.79 multiply(multiply(W, inverse(W)), inverse(multiply(inverse(multiply(inverse(multiply(W, inverse(W))), Y)), multiply(inverse(V), multiply(V, multiply(U, inverse(multiply(multiply(multiply(multiply(inverse(multiply(W, inverse(W))), multiply(multiply(W, inverse(W)), inverse(multiply(W, inverse(W))))), inverse(multiply(S, inverse(S)))), T), inverse(multiply(inverse(U), T))))))))))
% 2.06/0.79 = { by lemma 27 }
% 2.06/0.79 multiply(multiply(W, inverse(W)), inverse(multiply(inverse(multiply(inverse(multiply(W, inverse(W))), Y)), multiply(inverse(V), multiply(V, multiply(U, inverse(multiply(multiply(inverse(multiply(multiply(X2, inverse(X2)), inverse(multiply(inverse(multiply(W, inverse(W))), inverse(inverse(multiply(W, inverse(W)))))))), T), inverse(multiply(inverse(U), T))))))))))
% 2.06/0.79 = { by lemma 5 }
% 2.06/0.79 multiply(multiply(W, inverse(W)), inverse(multiply(inverse(multiply(inverse(multiply(W, inverse(W))), Y)), multiply(multiply(X2, inverse(X2)), inverse(multiply(inverse(multiply(W, inverse(W))), inverse(inverse(multiply(W, inverse(W))))))))))
% 2.06/0.79 = { by lemma 10 R->L }
% 2.06/0.79 multiply(multiply(W, inverse(W)), inverse(multiply(inverse(multiply(inverse(multiply(W, inverse(W))), Y)), multiply(multiply(inverse(multiply(W, inverse(W))), Y), inverse(multiply(inverse(multiply(W, inverse(W))), Y))))))
% 2.06/0.79 = { by lemma 18 }
% 2.06/0.79 multiply(multiply(W, inverse(W)), inverse(inverse(inverse(inverse(multiply(inverse(multiply(W, inverse(W))), Y))))))
% 2.06/0.79 = { by lemma 29 }
% 2.06/0.79 multiply(multiply(W, inverse(W)), multiply(inverse(multiply(W, inverse(W))), Y))
% 2.06/0.79 = { by lemma 31 R->L }
% 2.06/0.79 multiply(inverse(inverse(multiply(W, inverse(W)))), multiply(inverse(multiply(W, inverse(W))), Y))
% 2.06/0.79 = { by lemma 23 }
% 2.06/0.79 inverse(inverse(Y))
% 2.06/0.79
% 2.06/0.79 Lemma 35: inverse(multiply(X, inverse(Y))) = multiply(Y, inverse(X)).
% 2.06/0.79 Proof:
% 2.06/0.79 inverse(multiply(X, inverse(Y)))
% 2.06/0.79 = { by lemma 31 R->L }
% 2.06/0.79 inverse(multiply(X, inverse(inverse(inverse(Y)))))
% 2.06/0.79 = { by lemma 31 R->L }
% 2.06/0.79 inverse(inverse(inverse(multiply(X, inverse(inverse(inverse(Y)))))))
% 2.06/0.79 = { by lemma 34 R->L }
% 2.06/0.79 multiply(multiply(Z, inverse(Z)), inverse(multiply(X, inverse(inverse(inverse(Y))))))
% 2.06/0.79 = { by lemma 10 R->L }
% 2.06/0.79 multiply(multiply(inverse(inverse(Y)), inverse(X)), inverse(multiply(X, inverse(X))))
% 2.06/0.79 = { by lemma 9 R->L }
% 2.06/0.79 multiply(multiply(inverse(inverse(Y)), inverse(X)), inverse(multiply(W, inverse(W))))
% 2.06/0.79 = { by lemma 31 }
% 2.06/0.79 multiply(multiply(Y, inverse(X)), inverse(multiply(W, inverse(W))))
% 2.06/0.79 = { by lemma 20 }
% 2.06/0.79 multiply(multiply(Y, inverse(X)), multiply(inverse(inverse(inverse(inverse(V)))), inverse(V)))
% 2.06/0.79 = { by lemma 25 }
% 2.06/0.79 multiply(multiply(Y, inverse(X)), multiply(U, inverse(U)))
% 2.06/0.79 = { by lemma 17 }
% 2.06/0.79 inverse(inverse(multiply(Y, inverse(X))))
% 2.06/0.79 = { by lemma 24 }
% 2.06/0.79 multiply(inverse(inverse(Y)), inverse(inverse(inverse(X))))
% 2.06/0.79 = { by lemma 31 }
% 2.06/0.79 multiply(Y, inverse(inverse(inverse(X))))
% 2.06/0.79 = { by lemma 31 }
% 2.06/0.79 multiply(Y, inverse(X))
% 2.06/0.79
% 2.06/0.79 Goal 1 (prove_these_axioms): tuple(multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1)) = tuple(a2, multiply(a3, multiply(b3, c3)), multiply(inverse(b1), b1)).
% 2.06/0.79 Proof:
% 2.06/0.79 tuple(multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1))
% 2.06/0.79 = { by lemma 32 }
% 2.06/0.79 tuple(multiply(multiply(X, inverse(X)), a2), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1))
% 2.06/0.79 = { by lemma 34 }
% 2.06/0.79 tuple(inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1))
% 2.06/0.79 = { by lemma 31 }
% 2.06/0.79 tuple(a2, multiply(multiply(a3, b3), c3), multiply(inverse(a1), a1))
% 2.06/0.79 = { by lemma 32 }
% 2.06/0.79 tuple(a2, multiply(multiply(a3, b3), c3), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 21 R->L }
% 2.06/0.79 tuple(a2, multiply(multiply(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))), b3), c3), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 33 R->L }
% 2.06/0.79 tuple(a2, multiply(multiply(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))), b3), multiply(inverse(b3), inverse(multiply(inverse(c3), inverse(b3))))), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 35 R->L }
% 2.06/0.79 tuple(a2, multiply(multiply(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))), b3), inverse(multiply(multiply(inverse(c3), inverse(b3)), inverse(inverse(b3))))), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 31 }
% 2.06/0.79 tuple(a2, multiply(multiply(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))), b3), inverse(multiply(multiply(inverse(c3), inverse(b3)), b3))), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 7 R->L }
% 2.06/0.79 tuple(a2, multiply(multiply(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))), inverse(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))))), inverse(multiply(multiply(inverse(c3), inverse(b3)), inverse(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))))))), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 34 }
% 2.06/0.79 tuple(a2, inverse(inverse(inverse(multiply(multiply(inverse(c3), inverse(b3)), inverse(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)))))))))), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 31 }
% 2.06/0.79 tuple(a2, inverse(multiply(multiply(inverse(c3), inverse(b3)), inverse(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z)))))))), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 35 }
% 2.06/0.79 tuple(a2, multiply(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))), inverse(multiply(inverse(c3), inverse(b3)))), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 35 }
% 2.06/0.79 tuple(a2, multiply(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))), multiply(b3, inverse(inverse(c3)))), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 31 }
% 2.06/0.79 tuple(a2, multiply(inverse(inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(Z)))), inverse(Z))))), multiply(b3, c3)), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 21 }
% 2.06/0.79 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(Y, inverse(Y)))
% 2.06/0.79 = { by lemma 32 R->L }
% 2.06/0.79 tuple(a2, multiply(a3, multiply(b3, c3)), multiply(inverse(b1), b1))
% 2.06/0.79 % SZS output end Proof
% 2.06/0.79
% 2.06/0.79 RESULT: Unsatisfiable (the axioms are contradictory).
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